Theorem (Doob’s Optional Sampling Theorem — Discrete)

Let $(X_{n},\mathcal{F}_{n})$ be a martingale sequence and let $S,T$ be stopping times with respect to the Natural Filtration, $\mathcal{F}_{n}^{X}$ and let $n\in\mathbb{N}$ such that $S\le T\le n$ then

1. Integrable: $$X_{T},X_{S}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$$
2. Martingale Property: $$E[X_{T}|\mathcal{F}_{S}]=X_{S}\text{ a.s.}$$

Theorem (Doob’s Optional Sampling Theorem — Continuous)

Let $(X_{t})_{t\ge 0}$ be a right continuous $(\mathcal{F}_{t})_{t\ge 0}$-martingale. Let $S,T$ be stopping times and $S\le T\le N$ for some $N\in\mathbb{N}$ then

1. Integrable: $$X_{T},X_{S}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$$
2. Martingale Property: $$E[X_{T}|\mathcal{F}_{S}]=X_{S}\text{ a.s.}$$